Frame to frame interpolation for high-dimensional data visualisation using the woylier package

The woylier package implements tour interpolation paths between frames using Givens rotations. This provides an alternative to the geodesic interpolation between planes currently available in the tourr package. Tours are used to visualise high-dimensional data and models, to detect clustering, anomalies and non-linear relationships. Frame-to-frame interpolation can be useful for projection pursuit guided tours when the index is not rotationally invariant. It also provides a way to specifically reach a given target frame. We demonstrate the method for exploring non-linear relationships between currency cross-rates.

Zoljargal Batsaikhan https://github.com/zolabat (Monash University) , Dianne Cook http://www.dicook.org (Monash University) , Ursula Laa https://uschilaa.github.io (University of Natural Resources and Life Sciences Vienna)
2023-09-26

1 Introduction

When data has up to three variables, visualization is relatively intuitive, while with more than three variables, we face the challenge of visualizing high dimensions on 2D displays. This issue was tackled by the grand tour (Asimov 1985) which can be used to view data in more than three dimensions using linear projections. It is based on the idea of rotations of a lower-dimensional projection in high-dimensional space. The grand tour allows users to see dynamic low-dimensional (typically 2D) projections of higher dimensional space. Originally, Asimov’s grand tour presents the viewer with an automatic movie of projections with no user control. Since then new work has added interactivity to the tour, giving more control to users (Buja et al. 2005). New variations include the manual (Cook and Buja 1997) or radial tour (Laa et al. 2023), little tour, guided tour (Cook et al. 1995), local tour, and planned tour. These are different ways of selecting the sequence of projection bases for the tour, for an overview see Lee et al. (2022).

The guided tour combines projection pursuit with the grand tour and it is implemented in the tourr package (Wickham et al. 2011). Projection pursuit is a procedure used to locate the projection of high-to-low dimensional space that should expose the most interesting feature of data, originally proposed in Kruskal (1969). It involves defining a criterion of interest, a numerical objective function that indicates the interestingness of each projection, and an optimization for selecting planes with increasing values of the function. In the literature, a number of such criteria have been developed based on clustering, spread, and outliers.

A tour path is a sequence of projections and we use an interpolation to produce small steps simulating a smooth movement. The current implementation of tour in the tourr package uses geodesic interpolation between planes. The geodesic interpolation path is the shortest path between planes with no within-plane spin (see Buja et al. (2004) for more details). As a result, the rendered target plane could be a within-plane rotation of the target plane originally specified. This is not a problem when the structure we are looking for can be identified from any rotation. However, even simple associations in 2D, such as the calculated correlation between variables, can be very different when the basis is rotated.

Most projection pursuit indexes, particularly those provided by in tourr are rotationally invariant. However, there are some where the orientation of frames does matter. One example is the splines index proposed by Grimm (2016). The splines index computes a spline model for the two variables in a projection, in order to measure non-linear association. It compares the variance of the response variable to the variance of residuals, and the functional dependence is stronger when the index value is larger. It can be useful to detect non-linear relationships in high-dimensional data. However, its value will change substantially if the projection is rotated within the plane (Laa and Cook 2020). The procedure in Grimm (2016) was less affected by the orientation because it considered only pairs of variables, and it selects the maximum value found when exchanging which variable is considered as predictor and response variable.

Figure 1 illustrates the rotational invariance problem for a modified splines index, where we always consider the horizontal direction as the predictor variable, and the vertical direction as the response. Thus, our modified index computes the splines on one orientation, exaggerating the rotational variability. The example data was simulated to follow a sine curve and the modified splines index is calculated on different within-plane rotations of the data. Although they have the same structure, the index values vary greatly.

The lack of rotation invariance of the splines index raises complications in the optimization process in the projection-pursuit guided tour as available in tourr. Fixing this is the motivation of this work. The goal with the frame-to-frame interpolation is that optimization would find the best within-plane rotation, and thus appropriately optimize the index.

Three side-by-side scatterplots. The left side plot shows two variables, V5, V6, with a sine curve. The index value is the maximum of 1. The middle plot shows the two variables rotated 45 degrees clock-wise, and the calculated index value is 0.83. The right-side plot is rotated 60 degrees, and the calculated index value is 0.26.

Figure 1: The impact of rotation on a spline index that is NOT rotation invariant. The index value for different within-plane rotations take very different values: (a) original projection has maximum index value of 1.00, (b) axes rotated 45\(^o\) drops index value to 0.83, (c) axes rotated 60\(^o\) drops index to a very low 0.26. Geodesic interpolation between planes will have difficulty finding the maximum of an index like this because it is focused only on the projection plane, not the frame defining the plane.

A few alternatives to geodesic interpolation were proposed by Buja et al. (2005) including the decomposition of orthogonal matrices, Givens decomposition, and Householder decomposition. The purpose of the woylier package is to implement the Givens paths method in R. This algorithm adapts the Given’s matrix decomposition technique which allows the interpolation to be between frames rather than planes.

This article is structured as follows. The next section provides the theoretical framework of the Givens interpolation method followed by a section about the implementation in R. The method is applied to search for nonlinear associations between currency cross-rates.

2 Background

The tour method of visualization shows a movie that is an animated high-to-low dimensional data rotation. It is a one-parameter (time) family of static projections. Algorithms for such dynamic projections are based on the idea of smoothly interpolating a discrete sequence of projections (Buja et al. 2005).

The topic of this article is the construction of the paths of projections. The interpolation of these paths can be compared to connecting line segments that interpolate points in Euclidean space. Interpolation acts as a bridge between a continuous animation and the discrete choice of sequences of projections.

Interpolating paths of planes versus paths of frames

The tourr package implements geodesic interpolation between planes, and the final interpolation step will reach the rotation of the target frame that is avoiding any within-plane spin along the path. When the the orientation of projections matters interpolation between frames is required. The orientation of the frames could be important when a non-linear projection pursuit index function is used in the guided tour. This is illustrated by the different index values shown in the sketch in Figure 2, as well as the splines index for the sine curve in Figure 1.

XXX I like the dog sketch but currently seems not really needed - what would be nice would be showing a bit more here: starting plane is the dog looking in a different direction and also at an angle, target is the one aligned as currently seen in the front, geodesic path gets us to the target view at the angle as seen in the starting plane, givens path gives the exact target. Not sure how difficult it would be to make this though…

Plane to plane interpolation (left) and Frame to frame interpolation (right). We used dog index for illustration purposes. For some non-linear index orientation of data could affect the index.Plane to plane interpolation (left) and Frame to frame interpolation (right). We used dog index for illustration purposes. For some non-linear index orientation of data could affect the index.

Figure 2: Plane to plane interpolation (left) and Frame to frame interpolation (right). We used dog index for illustration purposes. For some non-linear index orientation of data could affect the index.

To describe the interpolation algorithms we will use the following notation.

Preprojection algorithm

In order to make the interpolation algorithm simple, we carry out a preprojection step to find the subspace that the interpolation path, \(F(t)\), is traversing. In other words, the preprojection step is defining the joint subspace of \(F_a\) and \(F_z\) and makes sure the interpolation path is limited to that space.

The procedure starts with forming an orthonormal basis by applying Gram-Schmidt to \(F_z\) with regards to \(F_a\), i.e. we find the \(p\times d\) matrix that contains the component of \(F_z\) that is orthogonal to \(F_a\). We denote this orthonormal basis by \(F_\star\). Then we build the preprojection basis \(B\) by combining \(F_a\) and \(F_\star\) as follows:

\[B = (F_a, F_{\star})\]

The dimension of the resulting orthonormal basis, \(B\), is \(p\times 2d\).

Then, we can express the original frames in terms of this basis:

\[F_a = B W_a, F_z = B W_z\]

The interpolation problem is then reduced to the construction of paths of frames \(W(t)\) that interpolate between the preprojected frames \(W_a\) and \(W_z\). By construction \(W_a\) is a \(2d\times d\) matrix of 1s and 0s. This is an important characteristic for our interpolation algorithm of choice, the Givens interpolation.

Givens interpolation path algorithm

A rotation matrix is a transformation matrix used to perform a rotation in Euclidean space. The matrix that rotates a 2D plane by an angle \(\theta\) looks like this:

\[ \begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix} \]

If the rotation is in the plane of two selected variables, it is called a Givens rotation. Let’s denote those 2 variables as \(i\) and \(j\). The Givens rotation is used for introducing zeros, for example when computing the QR decomposition of a matrix in linear algebra problems.

The interpolation method in the woylier package is based on the fact that in any vector of a matrix, one can zero out the \(i\)-th coordinate with a Givens rotation in the \((i, j)\)-plane for any \(j\neq i\) (Golub and Loan 1989). This rotation affects only coordinates \(i\) and \(j\) and leaves all other coordinates unchanged. Sequences of Givens rotations can map any orthonormal d-frame \(F\) in p-space to the standard d-frame \[E_d=((1, 0, 0, ...)^T, (0, 1, 0, ...)^T, ...).\]

The resulting interpolation path construction algorithm from starting frame \(F_a\) to target frame \(F_z\) is illustrated below. The example is for \(p=6\) and \(d=2\).

  1. Construct preprojection basis \(B\) by orthonormalizing \(F_z\) with regards tp \(F_a\) with Gram-Schmidt.

In our example, \(F_a\) and \(F_z\) are \(p\times d\) or \(6\times2\) matrices that are orthonormal. The preprojection basis \(B\) is \(p\times 2d\) matrix that is \(6\times 4\).

  1. Get the preprojected frames using the preprojection basis \(B\). \[W_a = B^TF_a = E_d\] and \[W_z = B^TF_z\]

In our example, \(W_a\) looks like:

\[ \begin{bmatrix}1 & 0 \\0 &1 \\ 0&0 \\0&0\end{bmatrix} \]

\(W_z\) is an orthonormal \(2d\times d\) matrix that looks like:

\[ \begin{bmatrix} a_{11} & a_{12} \\a_{21} &a_{22} \\ a_{31}&a_{32} \\a_{41}&a_{42}\end{bmatrix} \]

  1. Then, we can construct a sequence of Givens rotations that maps \(W_z\) to \(W_a\) with such angles that makes one element zero at a time:

\[ W_a = R_m(\theta_m) ... R_2(\theta_2)R_1(\theta_1)W_z\]

At each rotation, the angle \(\theta_i\) that zero out the next coordinate of a plane is calculated. Here \(m = \sum_{k=1}^d (2d - k)\), so when \(d=2\) we need \(m=5\) rotations with 5 different angles, each making one element 0. For example, the first rotation angle \(\theta_1\) is an angle in radians between \((1, 0)\) and \((a_{11}, a_{21})\). This rotation matrix would make element \(a_{21}\) zero:

\[R_1(\theta_1) = G(1, 2, \theta_1) = \begin{bmatrix} cos\theta_1 & -sin\theta_1 & 0 & 0 \\sin\theta_1 &cos\theta_1 & 0 &0 \\ 0&0&1&0 \\0&0&0&1\end{bmatrix}\] Here \(G(i,j,\theta_k)\) denotes a Givens rotation in components \(i\) and \(j\) by angle \(\theta_k\). In the same way we zero out the elements \(a_{31}\) and \(a_{41}\). Because of the orthonormality this means that now \(a_{11} = 1\) and that \(a_{12} = 0\). We thus need only two more rotations to zero out \(a_{32}\) and \(a_{42}\).

XXX something about how we calculate theta?

  1. The inverse mapping is obtained by reversing the sequence of rotations with the negative of the angles, we starts from the starting basis and end at the target basis.

\[R(\theta) = R_1(-\theta_1) ... R_m(-\theta_m), \ W_z = R(\theta)W_a\]

Performing these rotations would go from the starting frame to the target frame in one step. But we want to do it sequentially in a number of steps so interpolation between frames looks dynamic.

  1. Next we include the time parameter, \(t\), so that the interpolation process can be rendered in the movie-like sequence. We break each \(\theta_k\) into the number of steps, \(n_{step}\), that we want to take from the starting frame to the target frame, which means it moves by equal angle in each step. Here \(n_{step}\) should vary based on the angular distance between \(F_a\) and \(F_z\), such that when watching a sequence of interpolations we have a fixed angular speed.

  2. Finally, we reconstruct our original frames using \(B\). This reconstruction is done at each step of interpolation so that we have the interpolated path of frames as the result.

\[F_t = B W_t\]

At each time \(t\) we can project the data using the frame \(F_t\).

3 Implementation

We implemented each of the steps in the Givens interpolation path algorithm in separate functions and combined them into a single function givens_full_path() to produce the full set of \(F_t\). The same functions are used to integrate the Givens interpolation with the animate() functions of the tourr package. Table 1 lists the input and output of each functions and it’s descriptions, functions to use with animate() are described separately below.

Table 1: Primary functions in the woylier package.
name description input output
givens_full_path(Fa, Fz, nsteps) Construct full set of interpolated frames. Starting and target frame (Fa, Fz) and number of steps An array with nsteps matrices. Each matrix is an interpolated frame in between starting and target frames.
preprojection(Fa, Fz) Build the pre-projection space by orthonormalizing Fz with regard to Fa. Starting and target frame (Fa, Fz) B pre-projection p x 2d matrix
construct_preframe(F, B) Construct preprojected frames. An orthonormal frame F and the pre-projection matrix B Pre-projected frame in pre-projection space
row_rot(A, i, j, theta) Performs Givens rotation on the matrix A. Matrix A, components to rotate (i, j) and rotation angle theta Input matrix after Givens rotation
calculate_angles(Wa, Wz) Calculate angles of required rotations to map Wz to Wa. Pre-projected frames (Wa, Wz) Named list of angles
construct_moving_frame(Wt, B) Reconstruct interpolated frames using pre-projection. Pre-projection matrix B, frame of givens path A frame of a step in the interpolation

When using tourr we typically want to run a tour live, such that target selection and interpolation are interleaved, and the display will show the data for each frame \(F_t\) in the interpolation path. The implementation in tourr was described in Wickham et al. (2011), and with woylier we provide functions to use the Givens interpolation with the grand tour, guided tour and planned tour. To do this we rely primarily on the function givens_info() which calls the functions listed in the Table above and collects all necessary information for interpolating between a given starting and target frame. The function givens_path() then defines the interpolation and can be used instead of tourr::geodesic_path(). Wrapper functions for the different tour types are available to use this interpolation, since in the tourr::grand_tour() and other path functions this is fixed to use the geodesic interpolation. Calling a grand tour with Givens interpolation for direct animation will then use:

tourr::animate_xy(<data>, tour_path = woylier::grand_tour_givens())

4 Comparison of geodesic interpolation and Givens interpolation

The givens_full_path() function returns the intermediate interpolation step projections for a given number of steps. The code chunk below demonstrates the interpolation between 2 random bases in 5 steps.

givens_full_path(base1, base2, nsteps = 5)

To compare the path generated with the Givens interpolation to that found with geodesic interpolations we look at the rotation of the sine data shown in Figure 1. We consider a subset in \(p=4\) dimensions where the first two dimensions contain noise and the last two contain the sine curve. Starting from a random projection we want to interpolate towards the original sine curve. The path comparison is shown in Figure 3.

Four columns and five rows: first column projection coefficients from the Givens path; second column has scatterplots showing the corresponding projected data; third column projection coefficients from the geodesic path; fourth column has scatterplots showing the  corresponding projected data. First row is the same starting projection, subsequent rows are the steps for each path. Final row is the ending projection, which can be seen that both are the same plane, but the Givens goes to the specified frame showing the sine curve, and geodesic goes to a rotated view of the sine curve.

Figure 3: Comparison of Givens path and geodesic path between 2D projections. The Givens path preserves the frame ending at the provided basis (frame), while geodesic is agnostic to the particular basis. In general the geodesic is preferred because it removes within-plane spin, but occasionally it is helpful to very specifically arrive at the prescribed basis.

Plotting the interpolated paths

For further comparison and to check that the interpolation is moving in equally sized steps we directly plot the interpolated paths. The space of 1D projections defines a unit sphere, while 2D projections define a torus. To illustrate the space, points on the surface of the sphere and the torus shape are randomly generated by functions from the geozoo package (Schloerke 2016). The interpolated paths are then compared within that space.

A 1D projection of data in \(p\) dimensions corresponds to a linear combination where the weights are normalized. Therefore, we can plot the point on the surface of a hypersphere. In this case the Givens interpolation will reach the exact point, while geodesic interpolation might flip the direction and reach a point on the opposite side of the hypersphere. Figure 4 (left) shows the comparison of the interpolation steps using the same target plane, for an example with \(p=3\). Because the flipped target is close to the starting plane, the geodesic path is a lot shorter.

Two highlighted points on the surface of a 3D sphere (left) and 6D torus (right) connected by a path of points which are the interpolation steps, rotating.Two highlighted points on the surface of a 3D sphere (left) and 6D torus (right) connected by a path of points which are the interpolation steps, rotating.

Figure 4: Interpolation steps of 1D (left) and 2D (right) projections of 3D data made with a Givens path (forest green) and a geodesic path (deep red). The cream points represent the space of all projections, which is a sphere for 1D projections and a torus for 2D projections. In the 1D example, geodesic arrives at the opposite side of the sphere to Givens, indicating that it has flipped the direction of the vector in order to make the shortest path to the same plane. A similar thing happens for the 2D example, geosesic flips the sign of one basis vector, but it defines the same plane, as indicated by the cream circles.

In case of 2D projections, we can plot the interpolated path between 2 frames on a torus. A torus can be seen as crossing of 2 circles that are orthonormal, as is the case with our projection onto 2D. Figure 4 (right) compares the interpolation paths for \(p=3\).

For a 2D projection the same target plane is found when rotating the basis within the plane, or when reflecting across one of the two directions (the reflected basis can then also be rotated). This means the space of target bases is constrained to two circles on the torus, and these are disconnected because the reflection corresponds to a jump. In the high-dimensional space we can imagine the reflection as flipping over the target plane, resulting in a reflection of the normal vector on the plan. While the Givens interpolation will reach the exact basis, a geodesic interpolation towards the same target plane can land anywhere along those two circles, depending on the starting basis in the interpolation.

5 Application

This section describes the application of Givens interpolation path with guided tour to explore non-linear association in multivariate data. We use cross-rates for currencies relative to the US dollar. A cross-rate is an exchange rate between two currencies computed by reference to a third currency, usually the US dollar.

The data was extracted from openexchangerates and contains cross-rate for ARS, AUD, EUR, JPY, KRW, MYR between 2019-11-1 to 2020-03-31. Figure 5 shows how the currencies changed relative to USD over the time period. We see some collective behavior in March of 2020 with EUR and JPY increasing in a similar manner, and smaller currencies decreasing in value. This could be understood as a consequence of flight-to-quality at this uncertain times.

Six time series plots all very similar until March 2020 when JPY and EUR go up. All currencies go down in mid-March and JPY and EUR increase again at the end of March.

Figure 5: Examining the behaviour of six cross-currency rates (ARS, EUR, KRW, AUD, JPY, MYR) prior to and in the first month of the pandemic. All the currencies are standardised and the sign is flipped. JPY and EUR strengthend against the USD (high values) in March, while the other currencies weakened (low values).

We are interested in capturing this relations over time, and from the time series visualization we expect that we can capture the main dynamics in a two-dimensional projection from the six-dimensional space of currencies. Thus we start from \(p=6\) (the different currencies) and \(n=152\) the number of days in our sample. We expect that a projection onto \(d=2\) dimensions should capture the relation between the two groups of currencies mentioned above, and this should be identified within the noise of the random fluctuations.

We may expect that we can capture the dependence between the two groups using principal components analysis. Figure 6 shows a scatter plot matrix of the PCs of our dataset. Indeed we find strong non-linear association between the first two PCs. Investigation of the rotation shows that the first principal component is primarily a balanced combination between ARS, AUD, KRW and MYR (and a smaller contribution of EUR), and the second contribution is dominated by EUR and JPY contrasted with smaller contributions from ARS and MYR. Our next step is thus to use projection pursuit to identify the best projection matrix that captures the non-linear functional dependence.

PC1 and PC2 exihibt almost a sine wave, which starts with grey points and ends with blue. The other pairs of plots have little structure.

Figure 6: Scatterplot matrix of the six principal components. Observations in March 2020 are highlighted in dark blue, all other months are shown in grey. There is a strong non-linear dependence between PC1 and PC2.

Guided tour optimization

We now use the splines index to identify a projection with functional dependence between the first and second direction of the projection. Note here that because of strong linear correlations between the currencies, we start from the first four principal components, explaining over 97% of the variance in the data. To avoid starting from the view already identified in the first two principal components we start from a projections onto the third and forth principal components. The PCA display in tourr is used to show each projection axes display in terms of the original variables, while only touring within the smaller space spanned by the first four principal components.

The results are shown in Figure 7 and we see on the right that the guided tour with Givens interpolation has identified a non-linear functional dependence between the x and y axis in the final projection. The result on the left is using geodesic optimization, and the result in the middle is using a random search with geodesic interpolation. Since these methods do not allow for within-plane rotation they were not able to identify the same view even though they are using the same starting plane and index function.

Three animated plots showing biplot axes. The left and the middle scatterplot end at little structure, and the last scatterplot ends at a strong non-linear relationship.Three animated plots showing biplot axes. The left and the middle scatterplot end at little structure, and the last scatterplot ends at a strong non-linear relationship.Three animated plots showing biplot axes. The left and the middle scatterplot end at little structure, and the last scatterplot ends at a strong non-linear relationship.

Figure 7: optimization in the guided tour using geodesic optimization (left), simulated annealing with geodesic interpolation (middle) and simulated annealing with Givens interpolation (right). (Refresh page to re-start the animation.)

To understand the result in more detail we use tools from ferrn (Zhang et al. 2021) to show how the index value is changing over the optimization in Figure 8. Points indicate selected target planes, and are connected by lines showing the index value along the interpolated path between them.

The top row is showing the path found via geodesic optimization. This search strategy only moves along geodesic paths, and cannot select target frames with rotation within the plane. We can see that as a consequence the optimization converges to a local optimum and the search does not identify the pattern of interest.

In the middle we are showing the path found when using a random search in combination with geodesic interpolation. In this case the search can suggest any target frame (allowing for within-plane rotation), but the interpolation step will reset the orientation. This is why we see drops in the index value even from one target plane to the next, and the optimization also does not identify the pattern of interest.

Finally in the bottom row we see the result of the random search with Givens interpolation. While the index value can drop along an interpolation path, the value is strictly increasing for the target planes, as is required for the optimization. Note also that the search converged much faster (fewer target planes were selected). The length of the interpolated paths is similar, because for fixed angular step size the Givens interpolation will take more steps to interpolate between frames.

Three trace plots with points and connecting lines. Top plot is very smoothly increasing to maximum value, whereas the bottom two plots show up and down patterns.

Figure 8: Splines index value along the interpolated optimization path in the guided tour using geodesic optimization (top), simulated annealing with geodesic interpolation (middle) and simulated annealing with Givens interpolation (bottom). Points indicate index values at target planes selected during the optimization, and we see that with geodesic interpolation these values can decrease, impeding the optimization. XXX

6 Conclusion

The R package woylier provides implementation of Givens interpolation path algorithm that can be used as an alternative interpolation method for tour. The algorithm implemented in the woylier package comes from Buja et al. (2005). We illustrate the use of the functions provided in the package for R users.

The motivation to develop this package comes from rotational invariance problem of current geodesic interpolation algorithm implemeneted in tourr package. The package gives users the ability to detect non-linear association between variables more precisely.

It is important to mention that woylier package should be used in conjunction with the tourr package. The future improvements that needs to be done in the package is to generalize the interpolation for more than 2D projections of data.

CRAN packages used

tourr, woylier, geozoo, ferrn

CRAN Task Views implied by cited packages

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References

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Citation

For attribution, please cite this work as

Batsaikhan, et al., "Frame to frame interpolation for high-dimensional data visualisation using the woylier package", The R Journal, 2023

BibTeX citation

@article{woylier-article,
  author = {Batsaikhan, Zoljargal and Cook, Dianne and Laa, Ursula},
  title = {Frame to frame interpolation for high-dimensional data visualisation using the woylier package},
  journal = {The R Journal},
  year = {2023},
  issn = {2073-4859},
  pages = {1}
}